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If A is a symmetric matrix and \[{\text{n}} \in {\text{N}}\], write whether \[{{\text{A}}^{\text{n}}}\]is symmetric or skew-symmetric matrix or neither of these two.

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Hint: As we know that A is symmetric matrix, i.e. \[{\text{A = }}{{\text{A}}^{\text{T}}}\]. So, taking power of n of A and then satisfying it in the known condition we can lead to the solution of the given statement.

Complete step by step answer:

Given, A is a symmetric matrix and \[{\text{n}} \in {\text{N}}\],
As, A is a symmetric matrix, i.e. \[{\text{A = }}{{\text{A}}^{\text{T}}}\].
Now, take transpose of \[{{\text{A}}^{\text{n}}}\]
\[ \Rightarrow {{\text{(}}{{\text{A}}^{\text{n}}}{\text{)}}^{\text{T}}}\]
As we know that \[{{\text{(}}{{\text{A}}^{\text{n}}}{\text{)}}^{\text{T}}}{\text{ = (}}{{\text{A}}^{\text{T}}}{{\text{)}}^{\text{n}}}\]
\[ \Rightarrow {{\text{(}}{{\text{A}}^{\text{T}}}{\text{)}}^{\text{n}}}\]
As we know that \[{\text{A = }}{{\text{A}}^{\text{T}}}\]
So, \[{{\text{(}}{{\text{A}}^{\text{T}}}{\text{)}}^{\text{n}}} = {{\text{A}}^{\text{n}}}\]
Hence, \[{{\text{(}}{{\text{A}}^{\text{n}}}{\text{)}}^{\text{T}}}{\text{ = }}{{\text{A}}^{\text{n}}}\].
Hence, \[{{\text{A}}^{\text{n}}}\]is also a symmetric matrix.

Note: In mathematics, a matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. In mathematics, particularly in linear algebra, a skew-symmetric matrix is a square matrix whose transpose equals it’s negative.

Additional information: Matrices can be used to compactly write and work with multiple linear equations, referred to as a system of linear equations, simultaneously. Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps.